In probability theory, two events are said to be dependent if the probability of occurrence of one event is affected by the occurrence or non-occurrence of the other events. Informally speaking, this means that the occurrence of one event “depends” on the other. We now list out simple examples of dependent events in real life.
Some Examples of Dependent Events:
- Let A be the event that the weather forcast predicts rain and let B be the event that it actually rains outside. Then A and B are dependent events because if a reputed weather forcast predicts that it will rain then there is a very high chance of having rainy weather. Note that it is by no means certain that it will rain, even if the weather forecast predicts so. Also note that the dependence between the two events is not “causal”. The weather forecast does not “cause” the rain to fall.
- Suppose a box contains 2 balls – one black and one red. Suppose we randomly choose and remove balls from the box one at a time. Let A be the event that we pick a red ball at the first attempt and let B be the event that we pick a black ball at the second attempt. Then A and B are dependent events beacuse if A occurs and the red ball is removed then only the black ball remains in the box. Hence the black ball will 100% get picked in the second attempt. So the probability of occurrence of B is 100% depending on whether event A occurs.
- Suppose that a coin is tossed. Let A denote the event that we get a head and let B denote the event that we get a tail. Then A and B are clearly dependent beacuse the occurrence of A implies the non-occurrence of B and vice-versa.
- Studying thoroughly for an exam and obtaining a high score. Here the dependence is “causal” in the sense that studying hard for the exam definintely increases the chances of getting a high score.
- Doing weight training regularly and experiencing a gain in muscle mass. Once again, it is clear that if a person does regular weight training it leads to an increase in muscle mass. Hence the two events are dependent.
- Suppose that a dice is rolled. Let A be the event that we get a number strictly bigger than 2 and let B be the event that we get a prime number. Then if we A occurs we have two favourable possibilities for B because 3 and 5 are prime numbers. On the other hand, if A does not occur we have only one favourable outcome for B that is the prime number 2. Hence the probability of occurence of B is greater if A occurs.
- We finally look at a non-example. Suppose that on tossing a toin 10 times wehet heads continuously for all 10 trials. Is there a greater chance of getting tail in the eleventh toss? One might feel that since heads has occured many times there is a higher chance on getting tails. But this infact not true. The chances of getting a tail is still 50% because each coin toss is “independent” of the previous tosses. Hence successive coin tosses are not dependent.